American Institute of Mathematical Sciences

March  2021, 20(3): 1297-1317. doi: 10.3934/cpaa.2021021

Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations

 1 School of Mathematics, South China University of Technology, Guangzhou 510641, China 2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

* Corresponding author

Received  November 2020 Revised  December 2020 Published  March 2021 Early access  February 2021

Fund Project: Huancheng Yao and Changjiang Zhu were supported by the National Natural Science Foundation of China #11771150, 11831003, 11926346 and Guangdong Basic and Applied Basic Research Foundation #2020B1515310015. Haiyan Yin was supported by the National Natural Science Foundation of China #12071163, 11601165 and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University #ZQN-PY602

We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary $L^2$ energy methods.

Citation: Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
References:
 [1] M. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4nd edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. [2] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078. [3] R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4. [4] J. S. Fan and Y. X. Hu, Uniform existence of the 1-d complete equations for an electromagnetic fluid, J. Math. Anal. Appl., 419 (2014), 1-9.  doi: 10.1016/j.jmaa.2014.04.052. [5] J. S. Fan and Y. B. Ou, Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid, Nonlinear Anal., 106 (2014), 151-158.  doi: 10.1016/j.na.2014.04.018. [6] F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0. [7] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7. [8] F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914. [9] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014. [10] Y. T. Huang and H. X. Liu, Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system, Acta Math. Sci. Ser. B, 38 (2018), 857-888.  doi: 10.1016/S0252-9602(18)30789-6. [11] I. Imai, General Principles of Magneto-Fluid Dynamics. In: Magneto-Fulid Dynamics, Suppl. Prog. Theor. Phys., 24 (1962), 1-34. [12] S. Jiang and F. C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptot. Anal., 95 (2015), 161-185.  doi: 10.3233/ASY-151321. [13] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740. [14] S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869. [15] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. [16] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. Math. Sci., 62 (1986), 249-252. [17] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  doi: 10.21099/tkbjm/1496160397. [18] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 181-184. [19] T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328. [20] T. P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594.  doi: 10.1002/cpa.3160390502. [21] T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Commun. Math. Phys., 118 (1988), 451-465. [22] F. Q. Luo, H. C. Yao and C. J. Zhu, Stability of rarefaction wave for isentropic compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 59 (2021), 103234. doi: 10.1016/j.nonrwa.2020.103234. [23] T. Luo, H. Y. Yin and C. J. Zhu, Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system, Math. Models Methods Appl. Sci., 30 (2020), 343-385.  doi: 10.1142/S0218202520500098. [24] N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007. [25] A. Matsumura, Waves in compressible fluids: viscous shock, rarefaction, and contact waves, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2495–2548. doi: 10.1007/978-3-319-13344-7_60. [26] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [27] D. Mihalas and W. B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984. [28] I. S. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, 1962. [29] L. Z. Ruan, H. Y. Yin and C. J. Zhu, Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier-Stokes-Poisson system, Math. Methods Appl. Sci., 40 (2017), 2784-2810.  doi: 10.1002/mma.4198. [30] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [31] X. Xu, Asymptotic behavior of solutions to an electromagnetic fluid model, Z. Angew. Math. Phys., 69 (2018), 1-19.  doi: 10.1007/s00033-018-0945-6.

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References:
 [1] M. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4nd edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. [2] R. J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078. [3] R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4. [4] J. S. Fan and Y. X. Hu, Uniform existence of the 1-d complete equations for an electromagnetic fluid, J. Math. Anal. Appl., 419 (2014), 1-9.  doi: 10.1016/j.jmaa.2014.04.052. [5] J. S. Fan and Y. B. Ou, Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid, Nonlinear Anal., 106 (2014), 151-158.  doi: 10.1016/j.na.2014.04.018. [6] F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0. [7] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7. [8] F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914. [9] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014. [10] Y. T. Huang and H. X. Liu, Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system, Acta Math. Sci. Ser. B, 38 (2018), 857-888.  doi: 10.1016/S0252-9602(18)30789-6. [11] I. Imai, General Principles of Magneto-Fluid Dynamics. In: Magneto-Fulid Dynamics, Suppl. Prog. Theor. Phys., 24 (1962), 1-34. [12] S. Jiang and F. C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, Asymptot. Anal., 95 (2015), 161-185.  doi: 10.3233/ASY-151321. [13] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740. [14] S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869. [15] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. [16] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. Math. Sci., 62 (1986), 249-252. [17] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math., 10 (1986), 131-149.  doi: 10.21099/tkbjm/1496160397. [18] S. Kawashima and Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 181-184. [19] T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108.  doi: 10.1090/memo/0328. [20] T. P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594.  doi: 10.1002/cpa.3160390502. [21] T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Commun. Math. Phys., 118 (1988), 451-465. [22] F. Q. Luo, H. C. Yao and C. J. Zhu, Stability of rarefaction wave for isentropic compressible Navier-Stokes-Maxwell equations, Nonlinear Anal. Real World Appl., 59 (2021), 103234. doi: 10.1016/j.nonrwa.2020.103234. [23] T. Luo, H. Y. Yin and C. J. Zhu, Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system, Math. Models Methods Appl. Sci., 30 (2020), 343-385.  doi: 10.1142/S0218202520500098. [24] N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007. [25] A. Matsumura, Waves in compressible fluids: viscous shock, rarefaction, and contact waves, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 2495–2548. doi: 10.1007/978-3-319-13344-7_60. [26] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088. [27] D. Mihalas and W. B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984. [28] I. S. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, 1962. [29] L. Z. Ruan, H. Y. Yin and C. J. Zhu, Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier-Stokes-Poisson system, Math. Methods Appl. Sci., 40 (2017), 2784-2810.  doi: 10.1002/mma.4198. [30] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [31] X. Xu, Asymptotic behavior of solutions to an electromagnetic fluid model, Z. Angew. Math. Phys., 69 (2018), 1-19.  doi: 10.1007/s00033-018-0945-6.
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